Let p(y) denote the probability function associated with a

QUESTION:

Let \(p(y)\) denote the probability function associated with a Poisson random variable with mean \(\lambda\)
a. Show that the ratio of successive probabilities satisfies \(\frac{p(y)}{p(y-1)}=\frac{\lambda}{y}\), for \(y=1\), 2,....
b. For which values of  is \(p(y)>p(y-1)\)?

c. Notice that the result in part (a) implies that Poisson probabilities increase for awhile as

increases and decrease thereafter. Show that  \(p(y)\) maximized when  = the greatest integer

less than or equal to \(\lambda\).

Equation Transcription:

Text Transcription:

p(y)

lambda

p(y) over p(y-1)=lambda over y

y=1

p(y)>p(y-1)

p(y)

lambda